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OutputSensitive Algorithm for the EdgeWidth of an Embedded Graph
, 2010
"... Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest noncontractible and a shortest nonseparating cycle of G. If k is an integer, we can compute such a nontrivial cycle w ..."
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Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest noncontractible and a shortest nonseparating cycle of G. If k is an integer, we can compute such a nontrivial cycle with length at most k in O(gnk) time, or correctly report that no such cycle exists. In particular, on a fixed surface, we can test in linear time whether the edgewidth or facewidth of a graph is bounded from above by a constant. This also implies an outputsensitive algorithm to compute a shortest nontrivial cycle that runs in O(gnk) time, where k is the length of the cycle.
Asymptotic enumeration and limit laws for graphs of fixed genus
, 2010
"... It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface Sg of genus g grows asymptotically like c (g) n 5(g−1)/2−1 γ n n! where c (g)> 0, and γ ≈ 27.23 is the exponential growth rate of planar graphs. This generalizes the result for the planar ..."
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It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface Sg of genus g grows asymptotically like c (g) n 5(g−1)/2−1 γ n n! where c (g)> 0, and γ ≈ 27.23 is the exponential growth rate of planar graphs. This generalizes the result for the planar case g = 0, obtained by Giménez and Noy. An analogous result for nonorientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in Sg has a unique 2connected component of linear size with high probability.
ListColorCritical Graphs on a Fixed Surface
, 2009
"... A klistassignment for a graph G assigns to each vertex v of G a list L(v) ofadmissible colors, whereL(v)  ≥ k. A graph is klistcolorable (or kchoosable) if it can be properly colored from the lists for every klistassignment. We prove the following conjecture posed by Thomassen in 1994: “Ther ..."
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A klistassignment for a graph G assigns to each vertex v of G a list L(v) ofadmissible colors, whereL(v)  ≥ k. A graph is klistcolorable (or kchoosable) if it can be properly colored from the lists for every klistassignment. We prove the following conjecture posed by Thomassen in 1994: “There are only finitely many listcolorcritical graphs with all lists of cardinality at least 5 on any fixed surface. ” This generalizes the wellknown result of Thomassen on the usual graph coloring case. We use this theorem and specific parts of its proof to resolve the complexity status of the following problem about klistcoloring graphs on a fixed surface S, where k is a fixed positive integer. Input: A graph G embedded in the surface S. Question: Is G kchoosable? If not, provide a certificate (a listcolorcritical subgraph and the corresponding klistassignment). The cases k = 3, 4 are known to be NPhard (actually even Π p 2complete), and the cases k = 1, 2 are easy. Our main results imply that the problem is tractable for every k ≥ 5. In fact, together with our recent algorithmic result, we are able to solve it in linear time when k ≥ 5. Our proof yields even more: if the input graph is klistcolorable, then for any klistassignment L, we can construct an Lcoloring of G in linear time. This generalizes the wellknown lineartime algorithms for planar graphs by Nishizeki and Chiba (for 5coloring), and Thomassen (for 5listcoloring). We also give a polynomialtime algorithm to resolve the following question: Input: A graph G in the surface S, andaklistassignment L, wherek≥5.
Additive Approximation Algorithms for ListColoring MinorClosed Class of Graphs
"... It is known that computing the list chromatic number is harder than computing the chromatic number (assuming NP ̸ = coNP). In fact, the problem of deciding whether a given graph is flistcolorable for a function f: V → {c − 1, c} for c ≥ 3 is Π p 2complete. In general, it is believed that approxim ..."
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It is known that computing the list chromatic number is harder than computing the chromatic number (assuming NP ̸ = coNP). In fact, the problem of deciding whether a given graph is flistcolorable for a function f: V → {c − 1, c} for c ≥ 3 is Π p 2complete. In general, it is believed that approximating list coloring is hard for dense graphs. In this paper, we are interested in sparse graphs. More specifically, we deal with nontrivial minorclosed classes of graphs, i.e., graphs excluding some Kk minor. We refine the seminal structure theorem of Robertson and Seymour, and then give an additive approximation for listcoloring within k − 2 of the list chromatic number. This improves the previous multiplicative O(k)approximation algorithm [20]. Clearly our result also yields an additive approximation algorithm for graph coloring in a minorclosed graph class. This result may give better graph colorings than the previous multiplicative 2approximation algorithm for graph coloring in a minorclosed graph class [6]. Our structure theorem is of independent interest in the sense that it gives rise to a new insight on wellconnected Hminorfree graphs. In particular, this class of graphs can be easily decomposed into two parts so that one part has bounded treewidth and the other part is a disjoint union of boundedgenus graphs. Moreover, we can control the number of edges between the two parts. The proof method itself tells us how knowledge of a local structure can be used to gain a global structure, which gives new insight on how to decompose a graph with the help of localstructure information.
Choosability of Graphs with Bounded Order: Ohba’s Conjecture and Beyond
, 2013
"... c©Jonathan A. Noel, 2013 The choice number of a graph G, denoted ch(G), is the minimum integer k such that for any assignment of lists of size k to the vertices of G, there is a proper colouring of G such that every vertex is mapped to a colour in its list. For general graphs, the choice number is ..."
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c©Jonathan A. Noel, 2013 The choice number of a graph G, denoted ch(G), is the minimum integer k such that for any assignment of lists of size k to the vertices of G, there is a proper colouring of G such that every vertex is mapped to a colour in its list. For general graphs, the choice number is not bounded above by a function of the chromatic number. In this thesis, we prove a conjecture of Ohba which asserts that ch(G) = χ(G) whenever V (G)  ≤ 2χ(G) + 1. We also prove a strengthening of Ohba’s Conjecture which is best possible for graphs on at most 3χ(G) vertices, and pose several conjectures related to our work. ii Abrégé Le nombre de choix d’un graphe G, note ́ ch(G), est le plus petit entier k tel que pour toute affectation de listes de taille k au sommets de G, il y a une coloration de G tel que chaque sommet de G est colore ́ par une couleur de sa liste. En général, le nombre de choix n’est pas borne ́ supérieurement par une fonction du nombre chromatique. Dans cette thèse, nous démontrons une conjecture de Ohba qui affirme que ch(G) = χ(G) dès que V (G)  ≤ 2χ(G) + 1. Nous démontrons aussi une version plus forte de la conjecture de Ohba qui est optimale pour les graphes ayant au plus 3χ(G) sommets, et énonçons plusieurs conjectures par rapport a ̀ nos travaux. iii Declaration This thesis contains no material which has been accepted in whole, or in part, for any other degree or diploma. Chapters 4 and 6 of this thesis contain new contributions to knowledge. The results of these chapters have been, or will be, submitted for publication in peerreviewed journals. The result of Chapter 4 is based on joint work with Bruce A. Reed and Hehui Wu. The result of Chapter 6 is based on joint work with Douglas B. West, Hehui Wu, and Xuding Zhu. iv
Results on 5Coloring
"... Abstract In this paper, we provides the graph theoretic context of the results to follow. Here we give descriptions of the basic terminology and structures used for our results. also we explain how graphs can be embedded on surfaces other than the plane. More over we present an overview of the his ..."
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Abstract In this paper, we provides the graph theoretic context of the results to follow. Here we give descriptions of the basic terminology and structures used for our results. also we explain how graphs can be embedded on surfaces other than the plane. More over we present an overview of the history of coloring graphs on surfaces, especially in regards to 5coloring. Keywords orientable, 2connectedplanegraph, homeomorphic.